Ö4 = +2 or ±2

Real numbers

If we are working in real number system, we usually write 5 to stand for +5. In the same way we write Ö4 to stand for +Ö4 = +2.

Note that the situation is not the same in solving the equation:

x² = 4

x = ±Ö4 = ±2

and we write ± sign before the root sign.

Complex number system

Life is more harsh in complex numbers.

Put z = 4 = 4 + 0i = 4(cos 0 + i sin 0)

= 4(cos 2np + i sin 2np) , where n Î Z.

\ By de Moivres’ Theorem,

Obviously, when n = 0, we get the answer of 2(cos 0 + i sin 0) = +2.

when n = 1, we get the answer of 4(cos p + i sin p) = -2

Therefore,

Convention

As there is ambiguity in the square root process, some mathematicians reserves the index form for complex numbers and surds form for real numbers.

We write:

When someone asks you what is the square root of 4, ask him back what number system you are working. This may avoid unnecessary arguments.